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**Unformatted text preview: **1 5.4 The Fundamental Theorem of Calculus The derivatives of functions are defined as limits of difference quotients. Most of the derivative rules are derived using the limit definition. However, despite the fact that the definite integral is defined as a limit of Riemann sums, finding integrals as limits is often a very challenging task. Fortunately, the Fundamental Theorem of Calculus provides a much easier way of evaluating integrals via anti-differentiation. Example Consider a function defined as a definite integral: F ( x ) = Z x 2 xd x = 2 x x y f H x L = x The fact that F ( x ) = , which is the function we were integrating is not a coincidence. The following theorem provides the explanation: Theorem 4 The Fundamental Theorem of Calculus (Part 1) Let f ( x ) be a continuous function on the interval [ a,b ], then the function F ( x ) defined as: F ( x ) = Z x a f ( t ) dt is and and F ( x ) = d dx Z x a f ( t ) dt = Proof: 2 Example 1 Using the Fundamental Theorem of Calculus, find the derivatives of the...

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